Bone Heat Weighting Failed. No solution works
The solution is to scale up your object, parent, then scale back down (if you want to). I gave it a quick test with a single bone, which failed at default size, but worked after scaling both objects by a factor of 10.

The reason this works is because the automatic weighting system has a hard time with vertices that are very close together. Scaling up spreads everything out, making it easier to calculate.